The call to action:. Understanding addition and subtraction There is a lot of important work to be...

The call to action:

Understanding addition and subtraction• There is a lot of important work to be done to

ensure that students understand addition and subtraction in 1st and 2nd grade.

• The Call to Action is looking for units that connect understanding addition and subtraction to length.

Opportunity for coherence

MeasurementOperations and

Algebraic Thinking

Connecting the two domains together enhances students’ understanding of both.

• Linear measurement work in grade 1 and 2 asks students to consider the quantity of units required to represent length

• Students combine and compare lengths to deepen the understanding of the meaning of addition and subtraction

From the Draft K–5 Progression on Measurement and Data (measurement part)• Length and unit iteration are critical in understanding and using the number line in Grade 3 and beyond.

• Length is… one of the most prevalent metaphors for quantity and number, e.g., as the master metaphor for magnitude (e.g., vectors, see the Number and Quantity Progression)

https://commoncoretools.files.wordpress.com/2012/07/ccss_progression_gm_k5_2012_07_21.pdf

From the Draft K–5 Progression on Measurement and Data (measurement part)• To use a number line diagram to represent whole numbers as

lengths students need to understand that number lines have specific conventions• the use of a single position to represent a whole number and the use

of marks to indicate those positions. • a number line diagram is like a ruler in that consecutive whole

numbers are 1 unit apart, thus they need to consider the distances between positions and segments when identifying missing numbers

• Students think of a number line diagram as a measurement model and use strategies relating to distance, proximity of numbers, and reference points.

https://commoncoretools.files.wordpress.com/2012/07/ccss_progression_gm_k5_2012_07_21.pdf

By connecting measurement to addition and subtraction, students strengthen their understanding of a number line as iterations of equal size and of measurement as it relates to a quantity of units.

1st grade standards

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Do the math:

This task illustrates how using length units in addition and subtraction word problems connect students understanding of iterating units into work with operations. The context of the problem encourages students to use the number line, reinforcing the understanding of representing equal length, non-overlapping units.

https://www.illustrativemathematics.org/content-standards/tasks/196

2nd grade standards

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

This task illustrates the connection between addition and subtraction and moving equal units on the number line. The numbers students are working with are kept small to allow for focus on the connection between the operation and units of length.

https://www.illustrativemathematics.org/content-standards/tasks/1081

Think about this…

• We use the phrase “out of proportion” in all sorts of situations.

Like this:

Or this:

• 15 years in prison for stealing a gumball.

• Something in each was out of proportion. • What do we mean by that?

• We should be able to describe each situation in the same mathematically precise way.

• Hint: We need to focus on two things simultaneously, not one thing.• Proportionality is based on a relationship between

two quantities.

The Relevant Standards:

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Source: Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

Important elements:

From the Progressions • Ratios arise in situations in which two (or more)

quantities are related• In the Standards, a quantity involves measurement

of an attribute

http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

From the Progressions • Some authors distinguish ratios from rates, using

the term “ratio” when units are the same and “rate” when units are different; others use ratio to encompass both kinds of situations. • The Standards use ratio in the second sense,

applying it to situations in which units are the same as well as to situations in which units are different.

http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

More from the Progressions

• The quotient is sometimes called the value of the ratio 3 : 2. • In everyday language. the word “ratio” sometimes

refers to the value of a ratio• Ratios have associated rates.• The unit rate is the numerical part of the rate;• Equivalent ratios arise by multiplying each

measurement in a ratio pair by the same positive number.

http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

• Proportional relationships involve collections of pairs of measurements in equivalent ratios. • ratio notation should be distinct from fraction

notation• A collection of equivalent ratios can be graphed in

the coordinate plane. The graph represents a proportional relationship. • The unit rate appears in the equation and graph as

the slope of the line, and in the coordinate pair with first coordinate 1.

More from the Progressions

http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

Grade 6 (From the Progressions)

http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

• As students generate equivalent ratios and record them in tables, their attention should be drawn to the important role of multiplication and division in how entries are related to each other• In other words, when the elapsed time is divided

by 2, the distance traveled should also be divided by 2. More generally, if the elapsed time is multiplied (or divided) by N, the distance traveled should also be multiplied (or divided) by N

More from the Progressions

http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

• As students become comfortable with fractional and decimal entries in tables of quantities in equivalent ratios, they should learn to appreciate that unit rates are especially useful for finding entries

More from the Progressions

http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

Sidetrack:

• Sue and Julie were running equally fast around a track. Sue started first. When she had run 9 laps, Julie had run 3 laps. When Julie completed 15 laps, how many laps had Sue run?

Cramer, K., Post, T., & Currier, S. (1993). Learning and Teaching Ratio and Proportion: Research Implications. In D. Owens (Ed.), Research Ideas For the Classroom (pp. 159-178) NY: Macmillan Publishing Company. Downloaded 6/9/2011 from http://www.cehd.umn.edu/rationalnumberproject/93_4.html

Grade 7 (Progressions)

• They work with equations in two variables to represent and analyze proportional relationships.• Students recognize that graphs that are not lines

through the origin and tables in which there is not a constant ratio in the entries do not represent proportional relationships.• they write equations of the form y = cx, where c is

a constant of proportionality• unit rate as the amount of increase in y as x

increases by 1 unit in a ratio table [slope triangle]

http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

From the EQuIP Call to Action

Sample tasks:

https://www.illustrativemathematics.org/content-standards/6/RP/A/tasks/496

https://www.illustrativemathematics.org/content-standards/6/RP/A/tasks/496

https://www.illustrativemathematics.org/content-standards/6/RP/A/tasks/496

https://www.illustrativemathematics.org/content-standards/6/RP/A/3/tasks/1982

https://www.illustrativemathematics.org/content-standards/6/RP/A/3/tasks/1982

https://www.illustrativemathematics.org/content-standards/7/RP/A/2/tasks/181

• Task

A text book has the following definition for two quantities to be directly proportional:

We say that y is directly proportional to x if y=kx for some constant k .

For homework, students were asked to restate the definition in their own words and to give an example for the concept. Below are some of their answers. Discuss each statement and example. Translate the statements and examples into equations to help you decide if they are correct.

https://www.illustrativemathematics.org/content-standards/7/RP/A/2/tasks/1527

•Marcus: This means that both quantities are the same. When one increases the other increases by the same amount. An example of this would be the amount of air in a balloon and the volume of a balloon.•Sadie:Two quantities are proportional if one change is accompanied by a change in the other. For example the radius of a circle is proportional to the area. •Ben:When two quantities are directly proportional it means that if one quantity goes up by a certain percentage, the other quantity goes up by the same percentage as well. An example could be as gas prices go up in cost, food prices go up in cost.• Jessica:When two quantities are proportional, it means that as one quantity increases the other will also increase and the ratio of the quantities is the same for all values. An example could be the circumference of a circle and its diameter, the ratio of the values would equal π. https://www.illustrativemathematics.org/content-standards/7/RP/A/2/tasks/1527